Alexandroff pretopologies for structuring the digital plane

We explore the possibility of employing Alexandroff pretopologies as structures on the digital plane Z2 convenient for the study of geometric and topological properties of digital images. These pretopologies are known to be in one-to-one correspondence with reflexive binary relations so that graph-theoretic methods may be used when investigating them. We discuss such Alexandroff pretopologies on Z2 that possess a rich variety of digital Jordan curves obtained as circuits in a natural graph with the vertex set Z2. Of these pretopologies, we focus on the minimal ones and study their quotient pretopologies on Z2, which are shown to allow for various digital Jordan curve theorems. We also develop a method for identifying Jordan curves in the minimal pretopological spaces by using Jordan curves in their quotient spaces. Utilizing this method, we conclude the paper with proving a digital Jordan curve theorem for the minimal pretopologies.